\begin{conjecture} For all integers $k\geq1$ and $\ell\geq3$, there is an integer $f(k,\ell)$ such that for every set $P$ of at least $f(k,\ell)$ points in the plane, if each point in $P$ is assigned one of $k$ colours, then: \begin{itemize} \item $P$ contains $\ell$ collinear points, or \item $P$ contains a monochromatic line (that is, a maximal set of collinear points receiving the same colour) \end{itemize} \end{conjecture}

\begin{conjecture} For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear points or an empty hexagon. \end{conjecture}

Let $S$ be a set of points in the plane. Two points $v$ and $w$ in $S$ are \emph{visible} with respect to $S$ if the line segment between $v$ and $w$ contains no other point in $S$.

\begin{conjecture} For all integers $k,\ell\geq2$ there is an integer $n$ such that every set of at least $n$ points in the plane contains at least $\ell$ collinear points or $k$ pairwise visible points. \end{conjecture}